====== Timescale ======

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I don't know how to prove when the quasi-steady-state-approximation (QSSA) is valid.  But it is possible to check with a self-consistency condition:

We're exploring the case where one dynamical variable may be changing must faster than any others.  Let's use the example of \(x(t)\) where the dynamical equation for \(x\) is usually of order 2 or less in \(x\):

\[
  \diff{x}{t} = A x^2 + B x + C. 
\]

The prefactors \(A\), \(B\), and \(C\) may depend on the other dynamical variables but if our QSSA assumption is correct then the other variables should be slowly varying while \(x\) quickly equilibrates so we can treat them as invariant.

Under the QSSA assumption we can explicitly solve for \(x(t)\).  The general solution is

\[
  x(t) = \frac{-B + \sqrt{B^2 - 4 A C}}{2 A} \tan \left[ \frac{1}{2} \sqrt{B^2 - 4 A C} (t+k) \right] 
\]

where \(k\) is the integration constant.  Notice the factor \( \sqrt{B^2 - 4 A C} \) in the time-dependency.  That means we can rescale time to \(t'=t/\tau\) to get a scale-invariant time dependence where

\[
  \tau = 1/\sqrt{\abs{B^2 - 4 A C}} 
\]

is the natural timescale of the dynamics.

There are some degenerate cases to be aware of.  Here is the full set of solutions:

^  \(\diff{x}{t}\)  ^  \( x(t) \)  ^  \(1/\tau\)  ^
|  \( C \)  |  \( C t + k \)  |   \( \abs{C} \)  |
|  \( A x^2 \)  |   \( 1/(k-A t) \)  |   \( \abs{A} \)  |
|  \( A x^2 + B x + C \)  |  General \(x(t)\) above.  |   \( \sqrt{\abs{B^2 - 4 A C}} \)  |

